Here are some of my latest thoughts on category-theory… 
Graphs are to categories as lists are to monoids
Numbers are the lengths of lists which are the flattenings of trees which are the spannings of graphs. Unlike the first three, graphs have two underlying types of interest –the vertices and the edges– and it is getting to grips with this complexity that we attempt to tackle by considering their ‘algebraic’ counterpart: Categories.
In our exploration of what graphs could possibly be and their relationships to lists are, we shall mechanise, or implement, our claims since there will be many details and it is easy to make mistakes –moreover as a self-learning project, I'd feel more confident to make
Discovering Heyting Algebra
We attempt to motivate the structure of a Heyting Algebra by considering ‘inverse problems’.
For example,
- You have a secret number \(x\) and your friend has a secret number \(y\), which you've communicated to each other in person.
- You communicate a ‘message’ to each other by adding onto your secret number.
- Hence, if I receive a number \(z\), then I can undo the addition operation to find the ‘message’ \(m = z - y\).
What if we decided, for security, to change our protocol from using addition to using minimum. That is, we encode our message \(m\) as \(z = x ↓ m\). Since minimum is not invertible, we decide to send our encoded messages with a ‘context’ \(c\) as a pair \((z, c)\). From this pair, a unique number \(m′\) can be extracted, which is not necessarily the original \(m\). Read on, and perhaps you'll figure out which messages can be communicated 😉
This exploration demonstrates that relative pseudo-complements
- Are admitted by the usual naturals precisely when infinity is considered a number;
- Are exactly implication for the Booleans;
- Internalises implication for sets;
- Yield the largest complementary subgraph when considering subgraphs.
In some sense, the pseudo-complement is the “best approximate inverse” to forming meets, minima, intersections.
Along the way we develop a number of the theorems describing the relationships between different structural components of Heyting Algebras; most notably the internalisation of much of its own structure.
The article aims to be self-contained, however it may be helpful to look at this lattice cheat sheet (•̀ᴗ•́)و
Life & Computing Science by Musa Al-hassy is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License